Why is <n|n-2> equal to 0 in computing <n|\hat{a}^2|n>?

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SUMMARY

The discussion centers on the computation of the expression #### in quantum mechanics, specifically addressing why it equals zero. The conclusion is that this result arises from the orthogonality of Fock states, where ##=0## due to the properties of quantum harmonic oscillators. The use of square root terms, such as ##\sqrt{n}*\sqrt{n-1}##, reinforces the mathematical foundation of this result.

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Students and researchers in quantum mechanics, particularly those studying quantum states and operators, will benefit from this discussion.

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Homework Statement



This is not really homework. I'm just studying and came across this question. So, I'm not sure if I should post this question here on one of the physics sections.

On computing ##<n|\hat{a}^2|n>##, one arrives at ##\sqrt{n}*\sqrt{n-1}<n|n-2>=0## because ##<n|n-2>=0##

Now, why is that so?
 
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Isn't that just because the two Fock states are orthogonal?
 

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